Coupling of and with a strongly interacting Electroweak Symmetry Breaking Sector
We report the coupling of an external or state to a strongly interacting EWSBS satisfying unitarity. We exploit perturbation theory for those coupling of the external state, whereas the EWSBS is taken as strongly interacting. We use a modified version of the IAM unitarization procedure to model such a strongly interacting regime. The matrix elements , , , , and are all computed to NLO in perturbation theory with the Nonlinear Effective Field Theory of the EWSBS, within the Equivalence Theorem. This allows us to describe resonances of the electroweak sector that may be found at the LHC and their effect on other channels such as or where they may be discovered.
EPJ Web of Conferences \woctitleCONF12
The new boson with a mass of found at the LHC ATLAS (); CMS () would complete the Standard Model (SM) in its minimal version. The experimental collaborations at CERN are looking for deviations of its behaviour from that of the SM Higgs particle Aad:2013wqa (); Chatrchyan:2013lba (). The electroweak symmetry breaking happens at a scale of . New scalar resonances are constrained up to twophotons (). The constraint over new vector bosons reaches even higher in energy searches (). Thus, the study of the TeV scale is the goal of LHC Run-II. And it is expectable that new physics related with the Electroweak Symmetry Breaking Sector (EWSBS), if it exists, will be found at this scale.
To model the behaviour of a generic EWSBS with a theory which only includes the particles already known to exist there (the new Higgslike boson and the longitudinal components of gauge bosons , ), the so-called Higgs Effective Field Theory (HEFT) is used Alonso:2015fsp (); Alonso:2012px (); Contino:2013kra (); Gavela:2014uta (); Gavela:2014vra (); Buchalla:2015qju (); Buchalla:2013rka (); Feruglio:1992wf (); Contino:2010mh (); Bagger:1993zf (); Koulovassilopoulos:1993pw (); Burgess:1999ha (); Wang:2006im (); Grinstein:2007iv (); Brivio:2014pfa (); Rauch:2016pai (). A summary of these efforts can be found on the 4th CERN Yellow Report YellowReport (). This model is based on the old (Higgsless) Electroweak Chiral Lagrangian (ECL) Appelquist (), which is also inspired by Chiral Perturbation Theory (ChPt) for hadron physics ChPT ().
In our Refs. Delgado:2013loa (); Delgado:2013hxa (); Delgado:2014dxa (); Delgado:2014jda (); Delgado:2015kxa (), we simplify the study of the HEFT by means of the Equivalence Theorem ET (), which is valid for
Since the possible resonance region is above , this approximation is safe, and allows us to identify the longitudinal , with the pseudo-Goldstone bosons of symmetry breaking () in their scattering amplitudes. For instance,
We have found Delgado:2013loa (); Delgado:2013hxa (); Delgado:2014dxa (); Delgado:2014jda (); Delgado:2015kxa () that, for any parameter choice separating from the SM, the theory becomes strongly interacting at sufficiently high energy, and resonances may appear. Dispersion relations (as unitarization procedures) are used to deal with this non-perturbative regime. This also happens for the ChPT theory for hadron physics Truong:1988zp (); Dobado:1996ps (). In Ref. Dobado:1989gr (); GomezNicola:2001as (), unitarization procedures are tested for the hadron case of ChPT.
In this work, based on our recent works Delgado:2016rtd (); Castillo:2016erh (), we couple and states to the unitarized scattering amplitudes of our refs. Delgado:2014dxa (); Delgado:2015kxa (). The couplings between the EWSBS and states are perturbative in (negligible if compared with ) and , respectively. However, the EWSBS itself is strongly interacting (see Fig. 1). For the case, account must be taken that we are in the regime . This allows us to neglect diagrams like the 1-loop top mass renormalization, since they contribute to the scattering amplitudes with higher orders of . Note also that, in the high energy limit where the Equivalence Theorem can be applied, we can take consistently.
Note that we compute . and can both appear as initial and final states. Of course, and as final states are being tested at the LHC experiments, and physics of two photon decays has been pursued since the dawn of particle physics Yang:1950rg (). But we should not forget the physics of , since the photon can be a parton of the proton Manohar:2016nzj () or the electron in and colliders, respectively. The CMS collaboration Khachatryan:2016mud () is currently setting bounds to anomalous quartic gauge couplings by analysing . Photon colliders driven by lepton beams due to Compton backscattering could also become a future application of this work Telnov:2016lzw (); Gronberg:2014yfa ().
2 The Electroweak Chiral Lagrangian
First we quote the HEFT Lagrangian Delgado:2014jda (); Delgado:2015kxa (); Castillo:2016erh () which includes only the low-energy dynamics of the quark sector and the four light modes: three would-be Goldstone Bosons (WBGBs) and the Higgs-like particle . A global symmetry breaking pattern is considered.
The isospin can be parametrized by using the so-called spherical parametrization Delgado:2015kxa () by means of
Note that in the isospin basis, which is related to the charge one by and . The covariant derivative is given by
where the dots represent terms of higher order in .
The Higgs potential is expanded as
The SM is recovered for , , . Note that these terms are subleading in our approximation (Eq. 1), so that we could neglect the whole potential provided that this behaviour holds by whatever beyond SM theory happens to succeed. This is the case in most models of interest, and is a reasonable hypothesis since the constraints of these couplings have so far been found to be close to the SM values.
By expanding Eq. (3) without considering (yet) the Yukawa part, we obtain
where . We are not considering couplings with external transverse gauge bosons. Note that these neglected states do not appear in inner loops since this would be a higher order correction in .
Note that we have considered only couplings with the third quark generation (). This Lagrangian breaks custodial symmetry because of .
Since this is an EFT, in order to renormalize the tree level Lagrangian in Eq. (2) at the one-loop level, counterterms of dim. 8 are needed. For the EWSBS (strong) interactions, the minimal set of counterterms is (see Refs. Delgado:2013hxa (); Delgado:2015kxa ()),
And for the Yukawa coupling with top quarks Castillo:2016erh (),
3 Partial waves
The perturbative scattering amplitudes can be found in our Ref. Delgado:2015kxa (). For the representation of the states, we will use the isospin basis ( is the isospin and , its projection). Because of isospin symmetry, scattering amplitudes factorizes in this basis, and their values do not depend on . Unitarization procedures (based on dispersion relations) are most easily applicable over a partial wave decomposition Delgado:2015kxa () which, for states, are computed by means of
Since the state is an isospin singlet (which couples with ), if there is no mixing with channel, and the non-vanishing matrix elements are Delgado:2015kxa () . A similar expression to Eq. (14) is used for and ( in these case).
Note that partial waves of Eq. (14) have a chiral expansion
where corresponds to the LO term and , to the NLO computation and counterterms.
For the states, according to our Refs. Delgado:2014jda (); Delgado:2016rtd (), we have 4 polarization states that will be labeled as . The equation equivalent to Eq. (14), taking into account the effects of polarization, is
Note that only couples with 2- states, due to electric charge and angular momentum conservation. Indeed, parity conservation forbids 2- negative parity states coupling with . Thus, let us introduce the positive parity state . This is the only one that couples with states. Hence, we can define . For , the only non-vanising contributions come from the (positive parity states) and , so that . The perturbative scattering amplitudes themselves be found in Ref. Delgado:2014jda ().
Finally, the states only couple with and states Castillo:2016erh (). Even more, only couples with state. The corresponding partial waves are
and a similar expression for .
4 Unitarization procedures
As pointed out in Fig. 1, our key assumption will be a strongly interacting EWSBS while couplings with and states remain perturbative. The EWSBS scattering partial waves are unitarized by means of the same unitarization procedures (IAM, N/D, Improved-K matrix) that were exposed on our Refs. Delgado:2014dxa (); Delgado:2015kxa ().
We have two possibilites when unitarizing the EWSBS Delgado:2015kxa (). states couple with if and some of this conditions are verified: , or . Otherwise, the channel decouples. For instance, the single channel IAM is Delgado:2015kxa ()
Note that if coupling with channel happens, the coupled-channel versions of the unitarization procedures are required. Indeed, the strongly interacting regime could be triggered by even if is weak Delgado:2014dxa ().
Couplings with and are unitarized by means of a modification of the IAM and N/D procedures that uses as input the unitarized partial waves . This modification can be found in our Refs. Delgado:2016rtd (); Castillo:2016erh (). For the single-channel version of the IAM, it can be used
where , in this case, is the unitarized amplitude; , the corresponding perturbative one; , the unitarized ; and , the tree level matrix element . This guarantees that the phase of coincides with that of . For other channels, see Refs. Delgado:2014dxa (); Delgado:2015kxa ().
5 Numerical examples
On Figs. 2 and 3, a comparison between the , and can be found. We have used (see matrix elements in Refs. Delgado:2015kxa (); Delgado:2016rtd (); Castillo:2016erh ()), , , , and all the other NLO counterterms set to . This corresponds to the unitarization of the single channel plus weak couplings with and .
For computing the position of poles on the second Riemann sheet (resonances) in Fig. 3, we have used Cauchy’s Theorem via numerical integration, following the procedure that was already explained in our Ref. Delgado:2015kxa ().
Note the presence of a pole (broad resonance) on the second Riemann sheet in the same position in all the three channels (elastic scattering and couplings with and ). This was expected since the pole comes from the inner EWSBS (strongly coupled) dynamics, not from the physics of the (weak) coupling with and states.
Poles in the second Riemann sheet, if there are any, should come from the unitarization procedures applied to the inner dynamics of the EWSBS (Eq. 18 and other expressions from our Ref. Delgado:2015kxa ()). Couplings with and do not give rise to poles in the second Riemann sheet, according to Eq. (19) (and similar expressions in Refs. Delgado:2016rtd (); Castillo:2016erh ()), since perturbative partial waves (in particular, and in Eq. 19) are just polynomial in . Thus, the parameters associated with (, , , ) and (, , and ) should control the and coupling strength, but not the physics of the inner dynamics of the EWSBS. This is the picture which emerges from the chiral counting of Fig. 1. (However, if the parameters which control the couplings with and states were unnaturally large, either or would be non-perturbative and a full multi-channel unitarization involving should be carried out Castillo:2016erh ().)
We have computed the NLO scattering processes , and coupled them to a (hypothetical) strongly interacting EWSBS within the framework of HEFTs (). The Equivalence Theorem Delgado:2014jda (); Delgado:2014dxa (); Delgado:2015kxa () (which requires ) is used.
We have implemented all the relevant perturbative matrix elements and unitarization procedures for processes inside Fortran modules, following our Refs. Delgado:2014jda (); Delgado:2014dxa (); Delgado:2015kxa (); Delgado:2016rtd (); Castillo:2016erh (). Several unitarization procedures (based on dispersion relations) are used: IAM, N/D, Improve-K matrix and perturbative couplings with and states.
Our formalism assumes that the inner EWSBS dynamics (in the , and sector) is stronger than their electromagnetic coupling with s () and the electroweak couplings with (). Hence, any new resonance (within the range ) in channels would come from the inner EWSBS dynamics. This assumption could be broken for unnaturally large values of the NLO parameters related with and couplings. Such a situation would require the usage of a full coupled channel unitarization, including strongly or rescattering.
Even in the case that there are no new resonances, our scattering amplitudes can be a useful tool to parametrize separations from the SM in the regime . Currently, HEFTs themselves are being widely used in this way by CERN collaborations YellowReport ().
Finally, we are working within an expanded collaboration in the computation of simple estimates for collider cross-sections of typical resonances, and even for releasing a Monte Carlo module for the experimentalists.
We thank very useful conversation and suggestions from A. Castillo, Antonio Dobado, J.J. Sanz-Cillero, D. Espriu, and M.J. Herrero. We thank prof. Felipe J. Llanes-Estrada for reading the 1st version of the manuscript. Work supported by Spanish grants MINECO:FPA2011-27853-C02-01, MINECO:FPA2014-53375-C2-1-P and by BES-2012-056054.
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